Journal of Economic Dynamics & Control 28 (2004) 1781 – 1799
www.elsevier.com/locate/econbase
Dynamic Cournot-competitive harvesting of a
common pool resource
Leif K. Sandala , Stein I. Steinshamnb;∗
a Norwegian
b Institute
School of Economics and Business Administration, 5045 Bergen, Norway
for Research in Economics and Business Administration, 5045 Bergen, Norway
Received 30 August 2001; accepted 29 April 2003
Abstract
A variant of the classical harvesting game, where a number of competing agents simultaneously
harvest a common-pool natural resource, is analysed. The harvesting at each time is a Cournot
competition. The critical assumption made in the present analysis is that most (or all) individual
participants maintain a perspective which is wholly myopic. Three cases are analysed: (1) The
case of aggressive myopic Cournot competition, (2) co-operation and (3) competition when there
is an incumbent player. The development in the number of active participants is outlined, and
stability criteria for a dynamic Cournot-competitive game are given. The exact deadweight loss
due to lack of co-operation is calculated.
? 2003 Elsevier B.V. All rights reserved.
JEL classi!cation: C72; Q22
Keywords: Dynamic game-theory; Cournot competition; Natural resource exploitation
1. Introduction
This paper analyses a variant of the classical 9sheries harvesting game, where a
number of competing agents (9shing nations, :eets or vessels) simultaneously harvest
a common-pool 9sh stock, and do so over an extended period of time. The harvesting
at each time is a Cournot competition. That is to say, each nation, in choosing its
current harvest rate, takes into account that the current harvest-landings price depends
on the total harvest at the given time. Consequently landings price depends on the
simultaneous actions of all the nations. The focus is on the harvesting sector in each
∗
Corresponding author. Present address: SNF, Breiviksveien 40, 5045 Bergen, Norway. Tel.: +47-55 959
258; fax: +47-55 959 439.
E-mail address: [email protected] (S.I. Steinshamn).
0165-1889/$ - see front matter ? 2003 Elsevier B.V. All rights reserved.
doi:10.1016/j.jedc.2003.04.003
1782
L.K. Sandal, S.I. Steinshamn / Journal of Economic Dynamics & Control 28 (2004) 1781 – 1799
nation. In other words, Cournot competition here is similar to Cournot oligopoly with
asymmetric costs.
Each nation will also take into account its current cost of harvest, which depends on
the current 9sh stock biomass. However, the future stock biomass depends upon the
prior history of harvesting, and this circumstance links the evolution of future harvest
costs to current harvest rates and makes the model dynamic.
The critical assumption made in the present analysis is that most (or all) individual
participants in the 9shery maintain a perspective which is wholly myopic. That is,
in choosing its current harvest rate the myopic agent does not take into account this
impact of the current harvest on the future biomass trajectory. 1 Such myopic behavior
may profoundly aEect the long run evolution of the harvesting game. Myopic 9sheries
are also studied by HGamGalGainen et al. (1986, 1990) among others.
We believe that this circumstance of myopic decision making often characterises
traditional artesanal 9sheries, especially in developing countries. Very often there will
be no entity with the resources (or even the desire) to attempt a serious quanti9cation
of the relation between harvesting intensity and subsequent 9sh-stock dynamics. If this
relation is indeed signi9cant, but is ignored, then the ongoing 9shery may be seriously
sub-optimal. Ignoring the speci9c biological characteristics of stock demography can
have profound implications for the evolution of the 9shery, aEecting evolving stock
biomass and harvesting :eet structure, determining which agents will remain viable
participants in the ongoing 9shery and which will be eliminated in the continuing
competition. An important result in the paper is that we calculate the exact deadweight
loss due to competition compared to co-operation.
In our study we examine an idealised 9shery model, where the 9sh stock dynamics evolve according to a standard surplus production model, as conditioned by total
harvest. Three cases in particular will be analysed: (1) The case of aggressive myopic
Cournot competition, where all agents lack, or choose to ignore, information about the
dynamic stock demographics. This is typically the case in many 9sheries in developing
countries or for distant water 9shing nations. (2) Co-operation where the aim is to
maximise the total net revenue from the 9shery. (3) Competition where there is an
incumbent agent, for example a nation that is dedicated to keep the 9shery sustainable
and therefore takes account of the stock biomass growth function while the others still
do not; for example one coastal state versus a number of distant water 9shing nations.
Another example may be a stock that is shared between two nations where one nation
wants to take stock dynamics into account and do dynamic optimisation whereas the
other does not. A case in point may be Namibia and Angola; or even Norway and
Russia.
In each case all participants have full information about the market demand function, which determines the unit market price as a function of the total harvest. Each
participant also knows the current 9sh biomass level and the unit cost of harvesting,
as a function of biomass, for all of the other agents as well.
1 A relevant term for such agents in a continuous time model may be stroboscopic players. This would
be agents who control the system completely during very short time intervals.
L.K. Sandal, S.I. Steinshamn / Journal of Economic Dynamics & Control 28 (2004) 1781 – 1799
1783
As emphasised, an important feature of the model used here is that the ex vessel
landings price at any given time is a function of the combined harvest rate of all
participants. This circumstance, which introduces technical complications (due to nonlinearities in the objective functions) is treated only lightly in the 9sheries harvesting literature, although it is obviously an important aspect of real world 9sheries. To
our knowledge its previous considerations for a harvesting game is that of Takayama
and Simaan (1984), Reinganum and Stokey (1985), Dockner et al. (1989) and Datta
and Mirman (1999). Takayama and Simaan (1984) analyse a dynamic game involving two countries and a single resource. Reinganum and Stokey (1985) look at a
non-cooperative dynamic game involving n 9rms who are engaged in the extraction of
a non-renewable common property resource. Dockner et al. (1989) analysed a duopoly
situation with complete information, and Datta and Mirman (1999) use discrete dynamics to analyse the eEect of biological externalities when there are many countries
harvesting two 9sh stocks.
The outline of the present paper is as follows. The general model is presented in the
next section. Then we look at the three cases: (i) myopic competition, (ii) co-operation
and (iii) the case where one country is dedicated to maintain sustainability. Several
numerical examples are given. Finally the paper is concluded by a summary section.
2. General model
Assume that all agents are selling their harvest on the same market but they diEer
with respect to their eIciency through the cost function. Let x denote the common
9sh stock and hk denote agent k’s harvest. The total harvest of all agents is
h=
n
hk ;
k=1
where n is the number of active agents, 2 that is, by de9nition, agents with positive
harvest. It is assumed that there are no capacity constraints on individual agents. 3 This
is because emphasis is put on the eEects of competition, and capacity constraints will
reduce these eEects.
For simplicity, assume a linear market demand function which can be written on
inverse form as
p(h) = Q − qh;
where p(·) is the market price, and Q and q are market parameters. Each agent has
an individual cost function given as
k (hk ; x) = ck (x)hk ;
2
3
In the following n is always used to denote the number of active agents.
This assumption will be relaxed later.
1784
L.K. Sandal, S.I. Steinshamn / Journal of Economic Dynamics & Control 28 (2004) 1781 – 1799
where x is the current size of the common pool resource. This implies that agent k’s
net revenue, or utility function, in current value is given by
k = (Q − qh)hk − ck (x)hk
which can be rewritten
k = [pk (x) − qh]hk ;
hk ¿ 0;
(1)
where pk (x)=Q −ck (x) re:ects this agent’s eIciency. 4 The number of potential agents
is de9ned by the following:
Denition 1. The number of potential agents M = M (x) ¡ ∞ at some stock level x is
de9ned as all agents with pk ¿ 0.
The dynamics of the common stock are given by
dx
= f(x) − h;
dt
(2)
where f(x) is the biological growth function. A non-trivial steady-state exists if
h = f(x) for x ¿ 0. Criteria for stability of the steady state, if it exists, will be investigated later.
For a myopic agent, k, the objective is simply to maximise k at any point in time
given the prevailing stock x. In the case of an incumbent agent her objective is to
maximise 5
∞
e−k t k dt;
0
where k denotes agent k’s discount rate and t denotes time. Notice that diEerent
agents may have diEerent discount rates which may be quite realistic when the agents
are countries with diEerent economic conditions.
3. Myopic competition
In this section it is assumed that none of the agents use any of the information they
may have about the biological growth function. In other words, they do not perform
any dynamic optimisation due to the common pool characteristic of the 9shery. The
reason for such myopic behaviour may be fourfold. First, as long as there is more than
one agent, there is always an element of ‘the tragedy of the commons’. The rationale is:
whatever I do not harvest may be harvested by others and therefore I do not have any
incentives to save 9sh for tomorrow. Under such circumstances no one have incentives
to take the population dynamics of the stock into account. Second, the agents may
4
5
The results in this paper hold for a general x-dependence in Q and q.
Functional dependence of the variables is often skipped to improve the visual appearance of the equations.
L.K. Sandal, S.I. Steinshamn / Journal of Economic Dynamics & Control 28 (2004) 1781 – 1799
1785
simply not have, or do not believe in, information about population dynamics. The
biology of many 9sh stocks around the world is poorly understood and many have
not even been investigated. Even in the cases of 9sh stocks that have been thoroughly
studied, biologists from time to time admit that their models where wrong and the
stock dynamics has to be reconsidered from the beginning. Under such circumstances
it is no wonder that the possible participants in the 9shery do not believe strongly
in the biological models. Third, it is widely recognised that commercial 9shermen
often operate with short time horizons when they make their decisions. This has been
studied by Harms and Sylvia (2001). Fourth, with distant water :eets (DWF) roaming
the oceans seeking targets-of-opportunity just to reap oE any pro9ts that might exist
without interest in long-term conservation of the stock, these will have no incentives
to take the population dynamics into account.
The agents are ranked according to their economic eIciencies:
p1 (x) ¿ p2 (x) ¿ · · · ¿ pM (x)
and it is assumed, for simplicity, that the ranking remains constant at various stock
levels. Each agent will choose hk = 0 whenever pk − qh ¡ 0. As the agents do not
care about the dynamics, agent k chooses the most myopic decision rule which is to
maximise his k for all possible values of h−k , where h−k is de9ned as the other
agents’ harvest by h−k ≡ h − hk .
n
Denition 2. The general notation L−k = j=k Lj =nL−Lk is used, where : denotes
arithmetic mean. Further, L−k = (L−k )=(n − 1) is de9ned as the average of all the
other agents except k.
In order to maximise k agent k will solve
9k
= pk − qh − qhk = 0
9hk
(3)
which implies
pk
= h + hk ;
q
k = 1:: n;
(4)
without caring about the dynamics or the future. Expression (4) represents a system
of n equations and n unknowns, namely the hk s. By summing over all active agents
total harvest can be found:
h=
np
:
(n + 1)q
(5)
By substituting this into (4) again the harvest of one active agent appears as
hk =
npk − p−k
(n + 1)q
(6)
1786
L.K. Sandal, S.I. Steinshamn / Journal of Economic Dynamics & Control 28 (2004) 1781 – 1799
and the pro9t of one active agent is (from (4))
k = qh2k :
This is the Cournot-solution for the case where all agents have information about each
others eIciencies and about the market, but they do not take the biological growth
function into account.
The number of active agents can now be found by the following:
Proposition 1 (Activity principle). The number of active agents n is the maximal integer value of n 6 M satisfying npn ¿ p−n where M is the number of potential agents.
For all active agents pk ¿ ((n − 1)=n)p−k must apply.
The proof of Proposition 1 is straightforward. It is a direct consequence of the
de9nition of active, (5) and (6). When applying this proposition in practice we start
with the least eIcient of the potential agents and work our way down until the criterion
is satis9ed.
Proposition 1 has some quite interesting implications. Note, e.g., that if there are
only two potential agents, ((n − 1)=n) = 12 , and both agents are active unless one of
the agents is at least twice as eIcient as the other one. With three agents, ranked
according to eIciency, Agent 3 is active only if p3 ¿ (p1 + p2 )=3, and Agent 2 is
active only if p2 ¿ (p1 + p3 )=3. As n approaches in9nity, (n − 1)=n approaches one
from below. In other words, no matter how many agents there are, a suIcient criterion
to be active is that you are as eIcient as the average of the other active agents. This
may perhaps not be too surprising, but with few agents you can be much less eIcient
than that, and with only two agents, it is suIcient to be more than half as eIcient as
the other agent. Note that the number of active agents varies with the stock size, x, as
the parameters pk are functions of x. In other words, the number of active agents is a
function of the current stock size, n(x), at any point in time.
Proposition 1 implies that no active agent can be too far away from the average
of the other active agents. The following corollary derived from Proposition 1 will be
used later.
Corollary 1. No active agent is more than twice as e7cient as any of the other active
agents.
Proof. If such an agent exists, then Agent 1 must be more than twice as eIcient as
Agent n. From Proposition 1
npn ¿ p−n = p1 + p2 + · · · + pn−1 ¿ p1 + (n − 2)pn ⇒ 2pn ¿ p1 :
In other words, Agent 1 (the most eIcient) cannot be twice as eIcient as Agent
n (the least eIcient). This yields boundaries on the eIciency of one active agent
relative to the average of the other active agents. The boundaries are given by the next
corollary:
L.K. Sandal, S.I. Steinshamn / Journal of Economic Dynamics & Control 28 (2004) 1781 – 1799
1787
Corollary 2. The e7ciency of any of the active agents relative to the average e7ciency of the other active agents is bounded by
1−
1
pk
¡ yk ≡
¡ 2:
p−k
n
This corollary follows directly from Proposition 1. In the following yk will be referred to as k’s relative eIciency. The net revenue of one active agent is given by
2
2
1
n
npk − p−k
pk −
k = qh2k = q
=
p :
n+1
(n + 1)q
q
By summing over all active agents we get the total net revenue from the 9shery:
n
(n + 2)
2
2
2
p −
(7)
=q
hk =
n · p :
q
(n + 1)2
n
3.1. Dynamic aspect
Although none of the agents does any dynamic optimisation, the development of the
resource is nevertheless governed by Eq. (2). Using (5) this can be rewritten
dx
n · p
= F(x) ≡ f(x) −
:
dt
q(n + 1)
(8)
A steady state, x∗ , is characterised by h(x∗ ) = f(x∗ ). The stability of the steady state
depends upon the number of active agents and the biological and economic parameters.
This is summarised in the following proposition.
Proposition 2 (Local stability): A steady state, x∗ , is locally asymptotic stable if
1 + 1n q
f
p
+
¡ 0:
(9)
−
f
p
1 + 1n q
Proof. This proposition follows from linear stability, that is F(x∗ ) = 0 and
F (x∗ ) ¡ 0.
As the agents activities are based purely on economic criteria, the existence of a
steady state is not guaranteed. From (8) it is seen that a steady state exists whenever the
biological surplus production equals total myopic harvest. A steady state is guaranteed
if Agent 1’s cost structure is such that costs approach in9nity when the stock biomass
approaches zero; typical for trawl 9sheries. If n is constant, as it usually is close to
a steady state, this reduces to f =f ¡ p =p, see (9). The interpretation of this is
that the relative change in average eIciency must be greater than the relative change
in surplus production in a stable steady state.
The numerical example below is meant to illustrate the development of the stock
over time, and how the number of active agents depends upon the stock, when the
1788
L.K. Sandal, S.I. Steinshamn / Journal of Economic Dynamics & Control 28 (2004) 1781 – 1799
Fig. 1. Illustration of Example 1. Stock development over time with two diEerent initial stocks (0.6 and 0.1,
respectively). Steady state is 0.32.
eIciency parameters are highly sensitive to the stock size. This is typically the case
in search (bottom trawl) 9sheries.
Example 1. In this example costs depend strongly upon the stock. The agents are
characterised by pk = 3 − k(0:05=x2 ). The slope of the demand curve is q = 7, and the
surplus production function is f(x) = x(1 − x). The development of the stock and the
number of active agents are depicted in Figs. 1 and 2.
3.2. Monopoly
In this section we compare the competitive situation described above with the case
where one arbitrary active agent is given monopoly rights in harvesting, that is soleowner privileges. In particular, it is interesting to compare the net revenue of the total
competitive game with the net revenue of one agent with monopoly rights. This agent
is denoted by m. The following de9nitions have been used:
n
1 2
Am ≡ p−m ; Bm2 ≡ p2 −m =
pk ;
n−1
k=m
Cm2 ≡
Bm2
;
A2m
ym ≡
pm
1
∈ 1 − ;2 :
Am
n
Am is the arithmetic mean and Bm2 is the quadratic mean (root mean square) of the
other agents’ eIciencies. The term ym is Agent m’s eIciency relative to the average
L.K. Sandal, S.I. Steinshamn / Journal of Economic Dynamics & Control 28 (2004) 1781 – 1799
1789
Fig. 2. Illustration of Example 1. The number of players as a function of the stock. The steady-state stock
level is 0.32 and the steady-state number of players is three.
of the others, and the boundaries on ym are derived from Corollary 2. We also use the
relationships
np = [ym + n − 1]Am
np2 = [ym2 + (n − 1)Cm2 ]A2m :
With this notation the total net revenue from a 9shery with n agents can now be
rewritten
A2m
=
[(n2 + n − 1)ym2 − 2(n − 1)(n + 2)ym
(n + 1)2 q
+ (n − 1){(n + 1)2 Cm2 − (n − 1)(n + 2)}]:
(10)
By comparison, if one of the agents has a monopoly in the resource, her net revenue
will be
p2
A2
(11)
m = m = m · ym2 :
4q
4q
Whether one of the agents can generate more or less pro9ts than the competitive 9shery
depends on her relative eIciency ym . It turns out that the number of four active agents
becomes important. This can be stated more precisely as follows:
Proposition 3. For n 6 4
2n + 6
≡
yk ¡
3n + 5
is a su7cient criterion for an active agent, k, not to be suited as monopolist, i.e. not
to generate more pro!t alone than the competitive pool can do.
1790
L.K. Sandal, S.I. Steinshamn / Journal of Economic Dynamics & Control 28 (2004) 1781 – 1799
For n ¿ 4 this criterion will con:ict with the Activity principle (yk ¿ (n − 1)=n).
What this proposition says is that the eIciency of an active agent cannot be too far
below the average of the others if she shall be able to generate more net revenue as
a monopolist than the competitive 9shery can. The proposition is valid for any n ¿ 1,
but it is not binding for n ¿ 4. This is because the restriction given by the Activity
principle is stronger than the restriction above when there are more than four agents.
The proof can be sketched as follows: Assuming that (10) is no greater than (11) and
using the well-known 6 inequality Cm2 ¿ 1, implies
(3n2 + 2n − 5)ym2 − 8(n − 1)(n + 2)ym + 4(n − 1)(n + 3)
= (n − 1)(3n + 5)(ym − 2)(ym − ) 6 0 ⇒ ym ¿ :
In other words, the criterion yk ¡ is suIcient to state that agent k cannot produce
more net revenue as a monopolist than the pool. The criterion yk ¡ is impossible to
ful9l for more than 9ve agents. From Corollary 2 we have yk ¿ 1 − 1=n. A necessary
condition for Proposition 3 to be meaningful is that ¿ 1 − 1=n which implies n 6 4.
The two next examples show that Proposition 3 has non-trivial implications. In these
examples the pk s are independent of x. This is typically the case in schooling 9sheries.
Example 2. Four potential agents are characterised by [p1 ; p2 ; p3 ; p4 ] = [3; 2; 2; 1] and
q = 1. Only the 9rst three are active as Agent 1 is more than twice as eIcient as
Agent 4. Agent 3 is active as 3p3 = 6 ¿ p−3 = 5. From (6) we get h1 = 54 and h2 =
1
27
h3 = 14 and h = 74 . From (1) 1 = 25
16 and 2 = 3 = 16 and = 16 . As a monopolist
9
Agent 1 would generate 1 = 4 which is more than with competition whereas agents
2 and 3 would generate 2 = 3 = 1 which is less than with competition. This can be
seen directly from Proposition 3.
Example 3. Eight potential agents are characterised by [p1 ; p2 ; p3 ; p4 ; p5 ; p6 ; p7 ; p8 ] =
22
[1:25; 1:20; 1:15; 1:10; 1:05; 1; 1; 1] and q = 1. Now y = 28
31 ¿ = 29 , and hence no agents
can be excluded as unsuited as monopolists. The total pro9t with competition, =0:185
is less than what agent 8, the least eIcient one, would generate as a monopolist, which
is 8 = 0:25. From the Activity principle it is found that all eight are active agents.
It is seen from the last example that the propositions derived so far are not suIciently
strong to cover the case with many active agents. The next proposition, however, which
is based on a priori knowledge about the potential agents’ eIciencies only, give both
necessary and suIcient conditions for when an active agent is suited as monopolist.
Proposition 4. A necessary and su7cient condition for an active agent to generate
more net revenue than the total net revenue generated by the competitive pool (be
suited as a monopolist), is
1
and ym ¡ ym ¡ ym ;
Cm2 ¡ 1 +
3n + 5
6
See, e.g., Proposition 6.12 in Folland (1984).
L.K. Sandal, S.I. Steinshamn / Journal of Economic Dynamics & Control 28 (2004) 1781 – 1799
1791
where
ym =
4n + 8
− m ;
3n + 5
and the real value
m =
2n + 2
3n + 5
ym =
4n + 8
+ m
3n + 5
3n + 6 − (3n + 5)Cm2 :
If m is imaginary, then Cm2 ¿ 1+1=(3n+5), and the agent is unsuited as monopolist.
Note that Proposition 4 contains Proposition 3 as a special case as Cm2 ¿ 1 no matter
how many potential agents there are. For n ¿ 4 only Proposition 4 applies.
The proof for Proposition 4 follows directly from Eqs. (10) and (11). The diEerence
between the competitive pool’s pro9t and the monopolist’s pro9t is
2
(n − 1)(3n + 5)A2m
n+2
2
ym − 4
− m
≡ − m =
4(n + 1)2 q
3n + 5
which gives the proposition directly. Proposition 4 is applied in the next example. This
example is representative of a 9shery with many individual vessels of diEerent size,
for example some large purse seiners and many small coastal seiners.
Example 4. Consider a game with 1101 potential agents whose eIciencies are given
by pk = 1:1 − 0:001 · (k − 1). From the Activity Principle it is found that there are
46 active agents. From Proposition 4 [ym ; ym ] = [0:69; 1:99]. Hence all the 46 active
agents are suited as monopolists.
It is seen from the de9nitions that m decreases and ym increases with Cm2 as long
as they are real values. This means that there exists a value of Cm2 that makes ym = 1.
If Cm2 is greater than this value, then either ym ¿ 1 or m takes an imaginary value. In
both cases there will exist active agents who are unsuited as monopolists. By solving
the equation ym = 1, the following corollary is found:
Corollary 3. If an active agent, m, is less e7cient than the average of his competitors
and Cm2 ¿ 1 + n − 1=4(n + 1)2 , then this agent is unsuited as monopolist.
The above corollary represents a suIcient condition for unsuitability as monopolist
no matter how many agents there are.
It is relatively easy to construct examples that show that it under certain circumstances is possible even for an inactive agent to be suited as a monopolist.
Example 5. Let there be 8 potential agents. There are 7 equal agents with eIciencies
p1 = · · · = p7 = 1, and one agent with 20% lower eIciency, p8 = 0:8. Then by the
Activity Principle it is seen that this agent is inactive. However, by inserting into the
expressions for net revenue it is seen that agent 8 generates more revenue alone (as a
monopolist) than the competitive pool together.
1792
L.K. Sandal, S.I. Steinshamn / Journal of Economic Dynamics & Control 28 (2004) 1781 – 1799
4. Co-operation
In this section it is assumed that the agents co-operate, but they still act myopically.
In the case that the most eIcient agent has the capacity to take the total optimal
harvest, it will be left to this agent to do so, and the optimisation problem is similar to
the sole owner or monopolist referred to above. As Agent 1 is the most eIcient agent,
the optimal harvest is h = p1 =2q and the total net revenue is p12 =4q. This is a rather
trivial case, and the distribution of the net revenue is the only question that remains
to be negotiated.
In the case that the agents do have capacity constraints, they will 9ll up their capacities according to their eIciency. Let each agent’s 9xed capacity be denoted K1 ; K2 ; : : : :
Further, let us assume that n agents are needed, that is
h=
n
hn 6 Kn ;
hm ;
n ∈ {1; : : : ; M }:
m=1
The objective is to maximise
=
n
m =
m=1
n
(pm − qh)hm =
m=1
n
pm hm − qh2 :
m=1
The solution to this problem is to let the agents 9ll up their capacities according to
their eIciencies. Total harvest will then equal the sum of the capacities except for the
last active agent who may not 9ll up her capacity. The problem can be rewritten
max ();
h1 ; :::; hn
hm ∈ [0; Km ];
m ∈ {1; : : : ; n};
hk = 0 ⇒ hm = 0; m ¿ k;
hn = Kn ⇒ hm = Km ; m 6 n:
Let "n be used to denote the sum of the n 9rst capacities ("0 = 0). The solution
with respect to total harvest and the total number of agents can then be summarised
in the following proposition:
Proposition 5. In the case that the agents co-operate, but still act myopically, the
number of active agents, n ∈ {1; 2; 3; : : :}, is determined by the capacity constraints
pn
"n−1 ¡
(12)
¡ "n
2q
and total harvest is
pn
h=
:
2q
The double inequalities in (12) are mutually exclusive. If no solution exists, either
all potential agents are active with full capacity or there are no active agents at all.
Proposition 5 follows directly from Kuhn-Tucker’s suIcient conditions.
The deadweight loss due to myopic competition compared with myopic co-operation
is de9ned as the diEerence in producers’ surplus as we are ignoring the consumers’
L.K. Sandal, S.I. Steinshamn / Journal of Economic Dynamics & Control 28 (2004) 1781 – 1799
1793
surplus in this paper and concentrate on the harvesting sector. The distribution of the
surplus in the case of co-operation is not a topic in this paper.
The relative deadweight loss is de9ned as D ≡ 1 − comp =co-op where comp is net
revenue with competition and co-op is net revenue with co-operation. The deadweight
loss is highest in the case where capacity constraints are not binding, that is co-op = 1
and Agent 1 acts as a monopolist. This yields from (11) and (7)
2 2
p 1
n
p
+
− 1 ¿1 −
:
D = 1 − 4n ·
2
2
p1
p
(n + 1)
(n + 1)2
The right-hand side of the inequality (lower bound) is derived from noticing that
p=p1 ¿ 12 from Corollary 1 and p2 =p2 ¿ 1 from Cm2 ¿ 1, cf. footnote 5. An
interesting special case is when all active agents are equal, and in this case D = 1 −
(4n=(n + 1)2 ). The deadweight losses for 2, 3 and 4 active agents are 11%, 25% and
36%, respectively.
The deadweight loss referred to above is due to competition only. Capacity constraints in this context is a technological matter that contributes to reduce the deadweight loss.
5. An incumbent agent
This section looks at the case where one agent, m, is incumbent and performs dynamic optimisation whereas the others do not. In this case Agent m must take strategic
considerations into her decisions. Incumbent in this context implies that this agent takes
on an obligation to achieve a steady state if possible.
This may, e.g., be the case when there is one home-:eet and several possible ‘invaders’ who also have access to the resource. These invaders may represent distant
water :eets (DWF) who roam the oceans seeking targets-of-opportunity with intent to
reap the pro9ts wherever it occurs without interest in long-term viability of the stock.
As the DWFs come from far away they typically, but not necessarily, have higher
costs than the incumbent nation. This may either be higher 9xed costs associated with
moving the :eet or opportunity costs. It is assumed that all agents know each others
eIciencies, and that the DWFs have the same myopic utility functions as in the previous sections. It is further assumed that the incumbent agent, m, is one of the active
agents.
The other agents, apart from m, know that m is non-myopic, as this can be observed,
and Agent m knows that the others are myopic. Let us assume that there are n − 1
active myopic agents, and each of them does the same optimisation as earlier given
by (3). Adding all the active myopic agents yields
p pm
hm
np − pm − (n − 1)qh − q(h − hm ) = 0 ⇔ h =
−
+
:
(13)
q
nq
n
This represents total harvest as a function of the stock, x, for all values of hm =n. The
dynamic optimisation problem for the incumbent, non-myopic agent is now
∞
1
q 2
−m t
pm − p hm − hm dt
max
1+
e
hm
n
n
0
1794
L.K. Sandal, S.I. Steinshamn / Journal of Economic Dynamics & Control 28 (2004) 1781 – 1799
subject to
dx
p pm
hm
:
= f(x) −
+
−
dt
q
nq
n
In the section ‘Myopic competition’ it was shown that total harvest is given by
(5) when all behave myopically. If one of the agents behaves non-myopically, total
harvest is
h=
hm
p−m
+
:
nq
n
Depending on the eIciencies of the myopic agents, the non-myopic agent may be
faced with three diEerent situations:
Case 1: Steady state is not possible because the total harvest of the myopic agents
is greater than the surplus production, that is h−m ¿ f(x) for all x where f(x) ¿ 0.
Case 2: Steady state is not possible if all, including m, behave myopically. Steady
state may, however, be achieved if Agent m reduces her harvest or does not harvest
at all.
Case 3: Steady state is achieved even if all, including m, behave myopically.
Case 1 is rather trivial as the stock will inevitably go extinct, and the incumbent
agent may just as well play along and mine the resource in an optimal manner. This
case occurs if the sum of the other agents’ pk ’s is suIciently high. In Case 2 the
incumbent agent is obliged to provide a steady state. If her discount rate is low, this
may also be the optimal policy, and with zero discounting she will certainly bene9t
from a situation with a steady state. Case 3 is in many ways the most interesting case,
and this case occurs if the sum of the other agents’ pk ’s is suIciently small. In this
case the obligation to provide a steady state is not a binding constraint. Therefore, if
the incumbent agent deviates from myopic behaviour, it means that she is bene9tting
from it. This can be restated as follows:
Corollary 4. The incumbent agent bene!ts from non-myopic behaviour if there exists
a steady-state also when all agents, including the incumbent, behave myopically.
This corollary simply follows from the optimisation. The alternative may have been
optimal mining of the resource on the part of the incumbent, but this possibility is
excluded by the de9nition of incumbent. Let us now make the following de9nition:
Denition 3. Sustainable rent for the incumbent agent, S(x), is de9ned as Agent m’s
net revenue for any stock level as long as the stock does not change, in other words
when total harvest is equal to surplus production (h = f).
Agent m’s net price is then pm − qf ¿ 0 in order for m to be active. From (13)
hm = nf − (p−m )=q. Hence,
p−m
:
S(x) = (pm − qf) nf −
q
L.K. Sandal, S.I. Steinshamn / Journal of Economic Dynamics & Control 28 (2004) 1781 – 1799
1795
The sustainable rent is positive if and only if
pm
p−m
¡f¡
:
q
nq
This interval exists if Agent m is an active agent and not the only active. As S(x) is
continuous, it must also have a maximum in this interval. This leads to the following
proposition:
Proposition 6. In Case 3, Agent m’s sustainable rent, S(x), is greater than Agent m’s
corresponding myopic net revenue in the myopic steady state.
Proof. In the myopic steady state total harvest is given by (5) which must equal f(x).
Agent m’s net revenue is given by
m =
1
(pm − qf)2 :
q
Evaluating S(x) for the same stock level yields
1
(pm − qf)[nqf − p−m − (pm − qf)]
q
1
(pm − qf)f ¿ 0:
= (n + 1) 1 −
n
S(x) − m =
The implication of this proposition is that at the myopic steady state the incumbent
agent, m, can bene9t from deviating from myopic behaviour but still stay at the same
steady state. However, she can do even better than that by moving away from this
steady state and towards the optimal dynamic steady state.
Next we wish to 9nd the optimal dynamic behaviour of the incumbent agent. Following the approach outlined in Sandal and Steinshamn (2001) it is possible to 9nd
the optimal harvest for the incumbent agent as a feedback control law (function of
the state variable). If the incumbent has a discount rate of zero, the optimal feedback
harvest can be given explicitly as
p−m
hm (x) = max 0; nf −
− sign(x∗ − x) (S ∗ − S) · n=q ;
q
(14)
where S ∗ = S(x∗ ) = max S(x). The expression for the feedback rule follows from the
fact that the optimal Hamiltonian is constant with zero discounting. When applying this
rule it is imperative to keep in mind that n = n(x) is a step function determined by the
Activity Principle (Proposition 1). As we know that the maximum of S is an interior
solution, we can use the maximum principle to eliminate the costate variable and 9nd
the feedback rule given by (14). A proof of Catching-Up (CU) optimality in the case
1796
L.K. Sandal, S.I. Steinshamn / Journal of Economic Dynamics & Control 28 (2004) 1781 – 1799
of zero discounting, applying a version of Mangasarin’s SuIciency Theorem, 7 is given
in Sandal and Steinshamn (2001). This feedback rule will be used in the numerical
examples in the next section.
5.1. Numerical illustration of the case with an incumbent agent
In this section a numerical illustration of the case with one incumbent agent will
be given, and this will be compared to the case where all agents act myopically. It
is assumed that the discount rate is zero and that the incumbent agent has dedicated
herself to aim for a sustainable 9shery if possible. As long as the discount rate is
small compared to the intrinsic growth rate of the 9sh stock, discounting will only
have minor eEects on the optimal control, see Sandal and Steinshamn (1997).
In this example there are 9ve potential agents, and the numerical speci9cation is as
follows:
0:28
0:30
0:32
p1 = 1 −
; p2 = 1 −
; p3 = 1 −
;
x
x
x
0:35
0:40
; p5 = 1 −
; q=1
x
x
and the biological growth function is the rescaled logistic function
p4 = 1 −
f(x) = x(1 − x):
The rescaling implies that the carrying capacity is one, the intrinsic growth rate is one
and maximum sustainable yield is 0.25; see Clark (1990). In other words, the stock is
measured relative to the carrying capacity, and time units are measured relative to the
biological clock; i.e. as the inverse of the intrinsic growth rate.
Agent 1 is the incumbent, and it is interesting to compare the case where all 9ve
behave myopically with the case where Agent 1 behaves non-myopically. The development in the number of active agents when all behave myopically is illustrated in
Fig. 3.
Fig. 4 shows surplus production and the sole owner outcome with Agent 1 as the
sole owner, total harvest and Agent 1’s harvest when all behave myopically, and total
harvest and Agent 1’s harvest when Agent 1 is incumbent. Note that the steady states,
both when all are myopic and when Agent 1 is incumbent, are to the left of MSY
(maximum sustainable yield), whereas with a sole owner the optimal steady state is
always to the right of MSY when costs decrease with x and there is a reasonably small
discounting. An interesting, and somewhat counterintuitive, result is that the steady state
is characterised by a less conservative policy (lower standing stock) when Agent 1 is
incumbent than when all behave myopically. However, for lower stock levels the policy
when all behave myopically is less conservative than when Agent 1 is non-myopic and
behaves strategically.
Agent 1’s incumbent harvest is a stepwise function where the steps occur at the
entrance of new agents. At low stock levels the incumbent is alone and can bene9t
7
Theorem 13 in Seierstad and SydsTter (1987).
L.K. Sandal, S.I. Steinshamn / Journal of Economic Dynamics & Control 28 (2004) 1781 – 1799
1797
Fig. 3. The development in number of active players as a function of the stock when everybody behaves
myopically.
from a small harvest both in that she gets a high price and that the stock builds up fast
implying lower costs. As new agents enter, Agent 1 will increase her harvest in order
to meet the competition and reduce the harvest of the newcomer. This will reduce the
price but, at the same time, the build-up of the stock is slower which will postpone
the entrance of even more participants. Eventually the 9shery goes into a steady state,
in this example with three active agents. Fig. 4 also shows by how much Agent 1
increases her harvest each time a new agent enters, and it is seen that this is quite
substantial.
6. Summary and conclusions
In this paper, we have investigated a 9shery game where the market price depends
on total harvest and the participants act myopically in the sense that they do not take
stock dynamics into account in their decisions. The model has also been extended to
include the case where one participant takes the stock dynamics into account whereas
the others do not. The number of active participants out of the potential has been
endogenously determined. One of the main conclusions from this analysis is that both
the stock development and the number of active participants are heavily dependent upon
the individual eIciency parameters. With two agents one must be twice as eIcient as
the other in order to exclude the other. With a large number of agents, on the other
hand, one must be as eIcient as the average of the rest in order to be active. This
1798
L.K. Sandal, S.I. Steinshamn / Journal of Economic Dynamics & Control 28 (2004) 1781 – 1799
Fig. 4. Surplus production, sole owner’s harvest, total harvest when all are myopic, total harvest when Player
1 is incumbent, Player 1’s harvest when she is myopic, Player 1’s harvest when she is incumbent.
implies that the participants must have similar eIciencies in order to accommodate a
large number of active agents.
With one incumbent agent, the incumbent bene9ts from not being myopic if there
also exists a steady state when everybody behave myopically. However, the steady
state when all behave myopically may be less conservative (lower standing stock) than
when one of the participants is non-myopic.
Acknowledgements
We wish to thank Professor Robert W. McKelvey for the most helpful comments and
suggestions. Financial support from the Norwegian Research Council is acknowledged.
References
Clark, C.W., 1990. Mathematical Bioeconomics: The Optimal Management of Renewable Resources. Wiley,
New York.
L.K. Sandal, S.I. Steinshamn / Journal of Economic Dynamics & Control 28 (2004) 1781 – 1799
1799
Datta, M., Mirman, L.J., 1999. Externalities, market power and resource extraction. Journal of Environmental
Economics and Management 37, 233–255.
Dockner, E., Feichtinger, G., Mehlmann, A., 1989. Noncooperative solutions for a diEerential game model
of 9shery. Journal of Economic Dynamics and Control 13, 1–20.
Folland, G.B., 1984. Real Analysis: Modern Techniques and Their Applications. Wiley, New York.
HGamGalGainen, R.P., Ruusunen, J., Kaitala, V., 1986. Myopic Stackelberg equilibria and social coordination in
a share contract 9shery. Marine Resource Economics 3, 209–235.
HGamGalGainen, R.P., Ruusunen, J., Kaitala, V., 1990. Cartels and dynamic contracts in share9shing. Journal of
Environmental Economics and Management 19, 175–192.
Harms, J., Sylvia, G., 2001. A comparison of conservation perspectives between scientists, managers, and
industry in the West Coast ground9sh 9shery. Fisheries 26, 6–15.
Reinganum, J.F., Stokey, N.L., 1985. Oligopoly extraction of a common property natural resource: the
importance of the period of commitment in dynamic games. International Economic Review 26,
161–173.
Sandal, L.K., Steinshamn, S.I., 1997. Optimal steady states and the eEects of discounting. Marine Resource
Economics 12, 95–105.
Sandal, L.K., Steinshamn, S.I., 2001. A simpli9ed feedback approach to optimal resource management.
Natural Resource Modeling 14, 419–432.
Seierstad, A., SydsTter, K., 1987. Optimal Control Theory with Economic Applications. North-Holland,
Amsterdam.
Takayama, T., Simaan, M., 1984. DiEerential game theoretic policies for consumption regulation of renewable
resources. IEEE Transactions on Systems, Man and Cybernetics 14, 764–766.